Roots, Radicals, and Rational Exponents

Lesson 7: Roots, Radicals, and Rational Exponents

What is a root?

Labelling

Can you name the parts of a radical expression?Drag the label to the correct part of the image.

Drag labels to their correct position on the image

index

radical symbol

radicand

• Used to find roots

• A number/variable in front of the radical symbol is a coefficient, and thus multiplied to the radical expression

• No index? it is assumed to be 2.

Radical Expressions

A radical asks...

To answer those questions...

Knowing the first few perfect squares and cubes can be useful.

Let's list the first 10 perfect squares and perfect cubes.

Armed with the knowledge of

what a radical expression is and the list of perfect squares and perfect cubes, let's do a few examples.

Match

Match each of the following radical expressions to its equivalent expression.

• $$-\sqrt[]{121}$$

• $$-\sqrt[]{\frac{4}{25}}$$

• $$\sqrt[3]{343}$$

• $$\sqrt[3]{-125}$$

• $$\sqrt[3]{\frac{1}{27}}$$

• $$-11$$

• $$-\frac{2}{5}$$

• $$7$$

• $$-5$$

• $$\frac{1}{3}$$

What did all the previous expressions have in common?

What happens when we don't have a perfect square as a radicand?

Remember those perfect squares and cubes?!

• Look for the largest perfect square/cube factor of the radicand.

• Rewrite the expression using 2 radicals.

• Simplify the radical with the perfect square/cube radicand.

• If all radicands have no perfect square/cube factors, they are simplified.

• What do we do with the factors that aren't perfect squares/cubes?

Let's do this one together...

And one more...

Multiple Choice

Simplify $$4\ \sqrt[]{320}$$ .

• $$4\sqrt[]{40}$$

• $$32\sqrt[]{5}$$

• $$12\sqrt[]{5}$$

• $$8\sqrt[]{80}$$

Multiple Choice

Simplify $$3\sqrt[3]{108}$$ .

• $$9\sqrt[3]{4}$$

• $$6\sqrt[3]{4}$$

• $$3\sqrt[3]{12}$$

• $$18$$

Multiple Choice

Simplify: $$\sqrt{96}$$  

• $$16\sqrt{6}$$  
  

• $$6\sqrt{16}$$  

• $$9\sqrt{6}$$  

• $$4\sqrt{6}$$  

Multiple Choice

Simplify the following radical: $$\sqrt{32}$$  

• $$3\sqrt{2}$$  
  

• $$4\sqrt{8}$$  
  

• $$16\sqrt{2}$$  
  

• $$4\sqrt{2}$$  
  

Multiple Choice

Simplify:
$$\sqrt{100}$$  

• 1

• 10

• 10²

• Already simplified

Sometimes radical expression will contain variables:

• If the exponent is divisible by the index, divide.

• If the exponent is not divisible by the index, "break up" the exponent into a value divisible by the index and whatever remains
  

Let's do this one together...

And one more...

Multiple Choice

Simplify $$\sqrt[]{324a^3b^7}$$ .

• $$18ab^3$$

• $$18ab^3\sqrt[]{ab}$$

• $$18\sqrt[]{a^3b^7}$$

• $$324a^2b^6$$

Multiple Choice

Simplify

• $$x^2y\sqrt[]{xy}$$

• xy
• x²y²
• Can't Simplify

Multiple Choice

• A

• B

• C

• D

Multiple Choice

• A

• B

• C

• D

Multiple Choice

Find $$b$$ in $$22b^2=35$$ , where $$b$$ is a positive number. Isolate b by dividing the take the square root.

• $$\frac{35}{22}$$

• $$\left(\frac{35}{22}\right)^2$$

• $$\frac{\sqrt[]{35}}{22}$$

• $$\sqrt[]{\frac{22}{35}}$$

• $$\sqrt[]{\frac{35}{22}}$$

Multiple Choice

If $$x$$ is a real number, what is $$\sqrt[3]{x^{27}}$$ equivalent to? Divide the exponent by the index.

• $$x^{-9}$$

• $$x^{\frac{1}{9}}$$

• $$\left|x^9\right|$$

• $$x^9$$

• $$x^{24}$$

Multiple Choice

If a right triangle has legs of length 5x and x, which of the following expressions represents the length of its hypotenuse in terms of x?

a^2 + b^2 = c^2 (c is the hypotenuse)

Add like terms and simplify

• $$2x$$

• $$5x$$

• $$x\sqrt[]{6}$$

• $$2x\sqrt[]{6}$$

• $$x\sqrt[]{26}$$

Now let's talk about Rational Exponents

What does rational mean?
What does exponent mean?

Rational Exponents

Expressions with rational exponents can be rewritten as radicals using the following rules:

Refer to Desmos9.30 Slide 9

Easy way to remember...

Let's do this one together...

Write the expression in radical form. Simplify if needed.

And one more...

Write the expression in radical form. Simplify if needed.

Now let's write in exponential form...

Multiple Choice

Write the expression in radical form, simplifying if necessary:

$$7^{\frac{2}{3}}$$

• $$\sqrt[]{7^3}$$

• $$\sqrt[3]{49}$$

• $$7$$

• $$\sqrt[]{49}$$

Multiple Choice

Write with a rational exponent.

• 7³

• 7

• 7^3/4

• 7^4/3

Multiple Choice

Write the expression in radical form.

x^7/2

• $$\sqrt[]{x^7}$$

• $$7\sqrt[]{x^2}$$

• $$7x^2$$

• $$\sqrt[3]{x^7}$$

Multiple Choice

Write the following expression in exponential form:

• a.)

• b.)

• c.)

• d.)

Multiple Choice

Write the expression in exponential form.

$$\sqrt[3]{16}$$

• $$4$$

• $$16^{\frac{1}{3}}$$

• $$16^3$$

• $$16$$

Multiple Choice

Write with a rational exponent.

HINT: what root is it? It is "invisible"

• 216

• 6^1/2

• 6²

• 6^3/2

Multiple Choice

$$36^{\frac{1}{2}}\ =$$  

• 6
• $$\sqrt[]{6}$$

• $$18$$

• $$72$$

Multiple Choice

What is $$25^{\frac{1}{2}}-27^{\frac{1}{3}}$$ ?

• -4

• 2

• 8

• $$\frac{7}{6}$$

• $$\frac{52}{3}$$